What is the equation of the line through point \( (2,1) \) and perpendicular to the line through \( (-4,1) \) and \( (3,-2) ? \) \( \begin{array}{ll}\text { a) } y=\frac{7}{3} x+\frac{11}{3} & \text { go to station } 8 \\ \text { b) } y=\frac{3}{7} x+\frac{11}{3} & \text { go to station } 1 \\ \text { d) } y=-\frac{3}{7} x-\frac{11}{3} & \text { go to station } 10 \\ \text { d) } y=\frac{7}{3} x-\frac{11}{3} & \text { go to station } 5\end{array} \)

To find the equation of the line through the point \( (2,1) \) that is perpendicular to the line through the points \( (-4,1) \) and \( (3,-2) \), we will follow these steps: 1. **Find the slope of the line through the points \( (-4,1) \) and \( (3,-2) \)**: The slope \( m \) of a line through two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting the points \( (-4,1) \) and \( (3,-2) \): \[ m = \frac{-2 - 1}{3 - (-4)} = \frac{-3}{3 + 4} = \frac{-3}{7} \] 2. **Find the slope of the line perpendicular to this line**: The slope of a line that is perpendicular to another line is the negative reciprocal of the original slope. Therefore, if the slope of the original line is \( -\frac{3}{7} \), the slope \( m_{\perp} \) of the perpendicular line is: \[ m_{\perp} = -\frac{1}{-\frac{3}{7}} = \frac{7}{3} \] 3. **Use the point-slope form to find the equation of the line**: The point-slope form of a line is given by: \[ y - y_1 = m(x - x_1) \] where \( (x_1, y_1) \) is the point through which the line passes and \( m \) is the slope. Using the point \( (2,1) \) and the slope \( \frac{7}{3} \): \[ y - 1 = \frac{7}{3}(x - 2) \] 4. **Simplify the equation**: Distributing the slope: \[ y - 1 = \frac{7}{3}x - \frac{14}{3} \] Adding 1 (which is \( \frac{3}{3} \)) to both sides: \[ y = \frac{7}{3}x - \frac{14}{3} + \frac{3}{3} \] \[ y = \frac{7}{3}x - \frac{11}{3} \] Thus, the equation of the line through the point \( (2,1) \) and perpendicular to the line through \( (-4,1) \) and \( (3,-2) \) is: \[ y = \frac{7}{3}x - \frac{11}{3} \] The correct answer is: **d) \( y = \frac{7}{3} x - \frac{11}{3} \)**.